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We characterize the compactness of commutators in the Bloom setting. Namely,for a suitably non-degenerate Calder\'on--Zygmund operator $T$, and a pair ofweights $ \sigma , \omega \in A_p$, the commutator $ [T, b]$ is compact from $L ^{p} (\sigma ) \to L ^{p} (\omega )$ if and only if $ b \in VMO _{\nu }$,where $ \nu = (\sigma / \omega ) ^{1/p}$. This extends the work of the firstauthor, Holmes and Wick. The weighted $VMO$ spaces are different from theclassical $ VMO$ space. In dimension $ d =1$, compactly supported and smoothfunctions are dense in $ VMO _{\nu }$, but this need not hold in dimensions $ d\geq 2$. Moreover, the commutator in the product setting with respect to littleVMO space is also investigated.

Compactness of commutator of Riesz transforms in the two weight setting